What's Tested on the GRE: Geometry

Geometry topics include lines and angles, triangles—including isosceles, equilateral,and special right triangles—polygons, circles, multiple figures, three-dimensional figures (uniform solids), area, perimeter, and volume. You do not need to know how to do geometry proofs for the test. The geometry tested on the GRE is basic. There are only a few fundamental definitions and formulas you need to know. The GRE emphasizes new ways of applying these elementary rules.

An angle is formed by two lines or line segments which intersect at one point. The point of intersection is called the vertex. Angles are measured in either degrees or radians.
An acute angle is an angle whose measurement in degrees is between 0 and 90. A right angle is an angle whose measurement in degrees is exactly 90. An obtuse angle is an angle whose degree measure is between 90 and 180. A straight angle is an angle whose degree measure is exactly 180 degrees.

All of the angles on one side of a straight line sum to 180 degrees.
a + b + c = 180 degrees

All of the angles around one point must sum to 360 degrees.
a + b + c + d + e = 360 degrees
Perpendicular lines are formed when the angle between two lines is 90 degrees. The shortest distance from a point to a line is a line with a length such that the two lines form a 90 degree angle.
Two angles are supplementary if they share one line; i.e., if they sum of their angles is 180 degrees. Two angles are complementary if together they make a right angle; i.e., if the sum of their angles is 90 degrees.
To bisect an angle means to cut it in half. The two smaller angles will then have the same measurement.
If two parallel lines intersect with a third line, the third line is called a transversal. When this happens, all acute angles are equal and all obtuse angles are equal. Each acute angle is supplemental to each obtuse angle.

x and y are parallel lines, and z is the transversal
a = d = e = h
c = b = g = f
Vertical angles are a pair of opposite angles formed by intersecting lines. For the figure, a and d is an example of a pair of vertical angles. Vertical angles are equal.

Pay a lot of attention to diagrams. There can be a lot of information “hidden” in a diagram. If a diagram of an equilateral triangle gives you the length of one side, for instance, it actually gives the length of all sides. Similarly, if you are given the measure of one of the angles formed by the intersection of two lines, you can easily find the measure of all four angles. In fact, many geometry questions specifically test your ability to determine what additional information is implied by the information you are given in the diagram.
The diagrams provide basic information such as what kind of figure you are dealing with (is it a triangle? a quadrilateral?), the order of the points on lines, etc. However, the figures on the GRE are not drawn to scale unless otherwise stated. Because they are not drawn to scale,you can’t rely on how figures are drawn to conclude anything about the size of the figure. If a line looks straight in the diagram, you can assume it is straight. But you must be careful when using the diagram to judge relative lengths, angles, sizes, etc., since these may notbe drawn accurately. A square that looks twice as big as another is not necessarily twice as big. If an angle looks like a right angle, you cannot assume it is one, unless it is marked as such. If one side of a triangle looks longer than another, you cannot assume it is unless some other information tells you that it is. If a figure looks like a square, you don’t know it is a square; you only know it is a quadrilateral. This is especially important to bear in mind in the Quantitative Comparison section, where the diagram may lead you to believe that one column is greater, but logic will prove that you need more information.

For example, suppose triangle ABC has sides AC = 4 and BC = 5, but no right angle is marked or established by the problem. The Pythagorean theorem cannot be used simply because the triangle looks like a right triangle.

Instead, the triangle inequality determines the possible length of AB. A triangle’s third side must be greater than the difference and less than the sum of the other two sides:

5 − 4 < AB < 5 + 4

1 < AB < 9

If a Quantitative Comparison question compares AB with 6, the relationship cannot be determined. AB could be less than, equal to, or greater than 6.You can also, on occasion, use the diagram to your advantage by looking at the question logically. For instance:

We know from the question stem that we have a square and a circle, and we can see from the diagram that the circle is inscribed in the square—that is, it touches the square on all four sides. Whatever the area of the circle, we can see that the square’s area must be bigger; otherwise the circle wouldn’t fit inside it. So the right answer must be larger than the area of the circle, or larger than 4π. Now we can approximate 4π to a little more than 12. Since the correct answer is larger than 12, it must be either choice (D) or (E). And since you would expect to see π in the area of the circle, as it is here, that means π will not be in the area of the surrounding square. (The correct answer is 16.) This example highlights an important point: π appears very often on geometry problems, so you should have some idea of its value. It is approximately equal to 3.14, but for most purposes you only need remember that it

is slightly greater than 3.

Using Right Triangles on the GRE

The Pythagorean theorem relates the side lengths of a right triangle:

a² + b² = c²

The legs, a and b, meet at the right angle. The hypotenuse, c, is opposite the right angle and is always the longest side. Before applying the formula, test takers should confirm that the triangle has a right angle and identify which side is the hypotenuse.

For example, a right triangle with legs of 6 and 8 has a hypotenuse of 10:

6² + 8² = c²
36 + 64 = 100
c = 10

Recognizing Special Right Triangles

Two special right triangles have side-length relationships that can simplify GRE geometry calculations:

Triangle	How to recognize it	Side lengths
45°–45°–90°	A right triangle with two equal legs	x, x, x√2
30°–60°–90°	A right triangle whose hypotenuse is twice its shorter leg	x, x√3, 2x

For a 30°–60°–90° triangle, x is opposite the 30° angle, x√3 is opposite the 60° angle, and 2x is the hypotenuse.

These ratios apply only when the given information establishes the triangle’s type. A right triangle is not automatically one of these special triangles.

If a ratio is forgotten, the Pythagorean theorem can help reconstruct it. For example, a 45°–45°–90° triangle with legs of 1 has:

1² + 1² = c², so c = √2.

Solving GRE Geometry Problems One Step at a Time

A geometry problem may require several connected steps. When the final answer is not immediately clear, test takers can begin with a measurement they know and determine what it reveals next.

A useful approach is to:

• Identify the target: an area, angle, side length, perimeter, or volume.
• Label the diagram: record the given measurements and any values calculated along the way.
• Connect the shapes: look for shared sides, radii, diameters, or diagonals.
• Choose a relevant rule: use angle relationships for angles, area formulas for areas, and the Pythagorean theorem when a right triangle provides the needed side length.

Practice with a Square and Two Circles

A small circle is inscribed in a square, and the square is inscribed in a larger circle. If the small circle has an area of π, what is the area of the larger circle?

Step 1: Find the small circle’s radius.

Using A = πr²:

π = πr², so r = 1.

Step 2: Find the square’s side length.

The small circle’s diameter equals the square’s side length:

Side length = 2 × 1 = 2.

Step 3: Find the square’s diagonal.

The diagonal divides the square into two right triangles with legs of 2:

Diagonal² = 2² + 2² = 8

Diagonal = 2√2

Step 4: Find the larger circle’s area.

The square’s diagonal is the larger circle’s diameter, so the larger radius is √2:

Area = π(√2)² = 2π

Each calculation provides the information needed for the next step. During practice review, finding another measurement, such as the square’s area or either circle’s circumference, can help reinforce these relationships.